Limits and an Introduction to Calculus
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- Archibald Wilcox
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1 Chapter Sectio. Itroductio to Limits Limits ad a Itroductio to Calculus Objective: I this lesso ou leared how to estimate limits ad use properties ad operatios of limits. Course Number Istructor Date I. The Limit Cocept ad Defiitio of Limit (Pages ) Defie limit. If f () becomes arbitraril close to a uique umber L as approaches c from either side, the limit of f () as approaches c is L. This is writte as lim f () = L. c How to use the defiitio of a limit to estimate limits Describe how to estimate the limit lim umericall. Let f () = ( )/( + ). The costruct a table that shows values of f () whe is close to. Use the table to look for a umerical tred i the value of f () as approaches. This is a estimate of the limit. The eistece or oeistece of f () whe = c has o bearig o the eistece of... the limit of f () as approaches c. II. Limits That Fail to Eist (Pages ) The limit of f () as c does ot eist if a of the followig coditios is true:. f () approaches a differet umber from the right side of c tha it approaches from the left side of c. How to determie whether limits of fuctios eist. f () icreases or decreases without boud as approaches c. 3. f () oscillates betwee two fied values as approaches c. Larso/Hostetler/Edwards Precalculus with Limits: A Graphig Approach, Fifth Editio Studet Notetakig Guide IAE Copright Houghto Miffli Compa. All rights reserved. 75
2 76 Chapter Limits ad a Itroductio to Calculus Give a eample of a limit that does ot eist. Aswers will var. III. Properties of Limits ad Direct Substitutio (Pages ) Let b ad c be real umbers ad let be a positive iteger. Complete each of the followig properties of limits. How to use properties of limits ad direct substitutio to evaluate limits. lim b = b c. lim = c c 3. lim = c c 4. lim = c c, for eve ad c > 0 Let b ad c be real umbers, let be a positive iteger, ad let f ad g be fuctios with the followig limits. lim f ( ) = L ad lim g( ) = K c c Complete each of the followig statemets about operatios with limits.. Scalar multiple: lim[ b f ( )] = c bl. Sum or differece: lim[ f ( ) ± g( )] = c 3. Product: lim[ f ( ) g( )] = c L ± K LK f ( ) 4. Quotiet: lim = c g ( ) L/K, provided K 0 5. Power: lim [ f ( )] = L c Eample : Fid the limit: lim Larso/Hostetler/Edwards Precalculus with Limits: A Graphig Approach, Fifth Editio Studet Notetakig Guide IAE Copright Houghto Miffli Compa. All rights reserved.
3 Sectio. Itroductio to Limits 77 If p is a polomial fuctio ad c is a real umber, the lim p( ) = c p(c). If r is a ratioal fuctio give b r() = p()/q(), ad c is a real umber such that q(c) 0, the lim r( ) = c r(c) = p(c)/q(c). Eample : Fid the limit: 0 4 lim. Additioal otes Larso/Hostetler/Edwards Precalculus with Limits: A Graphig Approach, Fifth Editio Studet Notetakig Guide IAE Copright Houghto Miffli Compa. All rights reserved.
4 78 Chapter Limits ad a Itroductio to Calculus Additioal otes Homework Assigmet Page(s) Eercises Larso/Hostetler/Edwards Precalculus with Limits: A Graphig Approach, Fifth Editio Studet Notetakig Guide IAE Copright Houghto Miffli Compa. All rights reserved.
5 Sectio. Techiques for Evaluatig Limits 79 Sectio. Techiques for Evaluatig Limits Objective: I this lesso ou leared how to fid limits b direct substitutio ad b usig the dividig out ad ratioalizig techiques. Course Number Istructor Date I. Dividig Out Techique (Pages 79 79) The validit of the dividig out techique stems from... the fact that if two fuctios agree at all but a sigle umber c, the must have idetical limit behavior at = c. How to use the dividig out techique to evaluate limits of fuctios The dividig out techique should be applied ol whe... direct substitutio produces 0 i both the umerator ad the deomiator. A idetermiate form is... the fractio 0/0 that results whe direct substitutio produces 0 i both the umerator ad the deomiator. It has o meaig as a real umber ad is called a idetermiate form because it is ot possible to determie the limit from the form aloe. Whe ou ecouter a idetermiate form b direct substitutio ito a ratioal fuctio, ou ca coclude... that the umerator ad deomiator must have a commo factor. Eample : Fid the followig limit: lim II. Ratioalizig Techique (Page 793) Aother wa to fid the limits of some fuctios is to first ratioalize the umerator. This is called the ratioalizig techique, which meas multiplig the umerator ad deomiator b the cojugate of the umerator. How to use the ratioalizig techique to evaluate limits of fuctios Larso/Hostetler/Edwards Precalculus with Limits: A Graphig Approach, Fifth Editio Studet Notetakig Guide IAE Copright Houghto Miffli Compa. All rights reserved.
6 80 Chapter Limits ad a Itroductio to Calculus III. Usig Techolog (Pages ) To fid limits of oalgebraic fuctios,... ou ofte eed to use more sophisticated aaltic techiques such as a umerical solutio usig the table feature of a graphig utilit or a graphical solutio usig the zoom ad trace features of a graphig utilit. How to approimate limits of fuctios graphicall ad umericall IV. Oe-Sided Limits (Pages ) A oe-sided limit is... the limit at c of the fuctio f () as approaches c from either just the left or just the right. A limit from the left is deoted as lim f () = L. A limit from the c right is deoted as lim f () = L. c+ How to evaluate oesided limits of fuctios Eistece of a Limit If f is a fuctio ad c ad L are real umbers, the if ad ol if... equal to L. lim f ( ) = L c both the left ad right limits eist ad are V. A Limit from Calculus (Page 797) For a -value, the limit of a differece quotiet is a epressio of the form... How to evaluate limits of differece quotiets from calculus lim f ( + h) f (). h 0 h Direct substitutio ito the differece quotiet alwas produces the idetermiate form 0/0. Homework Assigmet Page(s) Eercises Larso/Hostetler/Edwards Precalculus with Limits: A Graphig Approach, Fifth Editio Studet Notetakig Guide IAE Copright Houghto Miffli Compa. All rights reserved.
7 Sectio.3 The Taget Lie Problem 8 Sectio.3 The Taget Lie Problem Objective: I this lesso ou leared how to approimate slopes of taget lies, use the limit defiitio of slope, ad use derivatives to fid slopes of graphs. Course Number Istructor Date I. Taget Lie to a Graph (Page 80) The taget lie to the graph of a fuctio f at a poit P(, ) is... the lie that best approimates the slope of the graph at the poit. How to defie the taget lie to a graph To determie the rate at which a graph rises or falls at a sigle poit,... ou ca fid the slope of the taget lie at that poit. II. Slope of a Graph (Page 80) To visuall approimate the slope of a graph at a poit,... draw the taget lie to the graph at the poit. The approimate the slope of the taget lie b estimatig the chage i the value of for each uit chage i. This ratio approimates the slope of the graph at the poit. How to use a taget lie to approimate the slope of a graph at a poit III. Slope ad the Limit Process (Pages ) A secat lie to a graph is... a lie through the poit of tagec ad a secod poit o the graph. How to use the limit defiitio of slope to fid eact slopes of graphs A differece quotiet is... the ratio [f( + h) f()]/h. Give the defiitio of the slope of a graph. The slope m of the graph of f at the poit (, f ()) is equal to the slope of its taget lie at (, f ()) ad is give b m = lim m sec = lim [f( + h) f()]/h, provided this limit eists. h 0 h 0 Larso/Hostetler/Edwards Precalculus with Limits: A Graphig Approach, Fifth Editio Studet Notetakig Guide IAE Copright Houghto Miffli Compa. All rights reserved.
8 8 Chapter Limits ad a Itroductio to Calculus Eample : Use the limit process to fid the slope of the graph of f ( ) = + 5 at the poit (3, ). This graph has a slope of 6 at the poit (3, ). IV. The Derivative of a Fuctio (Pages ) The derivative of f at is the fuctio derived from... the limit process to represet the slope of the graph of f at the poit (, f()). How to fid derivatives of fuctios ad use derivatives to fid slopes of graph Give the formal defiitio of the derivative. The derivative of f at is give b f () = lim [f( + h) f()]/h, h 0 provided this limit eists. The derivative f () is a formula for... the slope of the taget lie to the graph of f at the poit (, f()). Eample : Fid the derivative of f () = 4 f ( ) = 9. Homework Assigmet Page(s) Eercises Larso/Hostetler/Edwards Precalculus with Limits: A Graphig Approach, Fifth Editio Studet Notetakig Guide IAE Copright Houghto Miffli Compa. All rights reserved.
9 Sectio.4 Limits at Ifiit ad Limits of Sequeces 83 Sectio.4 Limits at Ifiit ad Limits of Sequeces Objective: I this lesso ou leared how to evaluate limits at ifiit ad fid limits of sequeces. Course Number Istructor Date I. Limits at Ifiit ad Horizotal Asmptotes (Pages 8 84) Defie limits at ifiit. How to evaluate limits of fuctios at ifiit If f is a fuctio ad L ad L are real umbers, the statemets lim f () = L ad lim f () = L deote the limits at ifiit. The first is read the limit of f () as approaches is L, ad the secod is read the limit of f () as approaches is L. To help evaluate limits at ifiit, ou ca use the followig: If r is a positive real umber, the lim r = 0. If r is defied whe < 0, the lim r = 0. Eample : Fid the limit: lim 3 3 If f () is a ratioal fuctio ad the limit of f is take as approaches or, Whe the degree of the umerator is less tha the degree of the deomiator, the limit is 0. Whe the degrees of the umerator ad the deomiator are equal, the limit is the ratio of the coefficiets of the highest-powered terms. Whe the degree of the umerator is greater tha the degree of the deomiator, the limit does ot eist. Larso/Hostetler/Edwards Precalculus with Limits: A Graphig Approach, Fifth Editio Studet Notetakig Guide IAE Copright Houghto Miffli Compa. All rights reserved.
10 84 Chapter Limits ad a Itroductio to Calculus II. Limits of Sequeces (Pages 85 86) For a sequece whose th term is a, as icreases without boud, if the terms of the sequece get closer ad closer to a particular value L, the the sequece is said to coverge to L. Otherwise, a sequece that does ot coverge is said to diverge. How to fid limits of sequeces Give the defiitio of the limit of a sequece. Let f be a fuctio of a real variable, such that lim f () = L. If {a } is a sequece such that f () = a for ever positive iteger, the lim a = L. Eample : Fid the limit of the sequece 4 a ( 3)(4 ) =. 4 3 Homework Assigmet Page(s) Eercises Larso/Hostetler/Edwards Precalculus with Limits: A Graphig Approach, Fifth Editio Studet Notetakig Guide IAE Copright Houghto Miffli Compa. All rights reserved.
11 Sectio.5 The Area Problem 85 Sectio.5 The Area Problem Objective: I this lesso ou leared how to fid limits of summatios ad use them to fid areas of regios bouded b graphs of fuctios. Course Number Istructor Date I. Limits of Summatios (Pages 80 8) The followig summatio formulas ad properties are used to evaluate fiite ad ifiite summatios. How to fid limits of summatios. c =. i = 3. i = 3 4. i = c ( + )/ ( + )( + )/6 ( + ) /4 5. ( ai ± bi ) = ai ± b 6. kai = k a i i To fid the limit of a summatio,... begi b applig summatio formulas ad properties to covert the summatio form to ratioal form. Oce i ratioal form, ou ca use the techiques from the previous sectio to fid the limit as. Eample : Fid the limit of S() as. S( ) = 0 i 5 3 Larso/Hostetler/Edwards Precalculus with Limits: A Graphig Approach, Fifth Editio Studet Notetakig Guide IAE Copright Houghto Miffli Compa. All rights reserved.
12 86 Chapter Limits ad a Itroductio to Calculus II. The Area Problem (Pages 83 85) Describe the area problem. The area problem is to fid the area of the regio R bouded b the graph of a oegative, cotiuous fuctio f, the -ais, ad the vertical lies = a ad = b. The eact area of a plae regio R is give b... limit of the sum of rectagles as approaches. the How to use rectagles to approimate areas of plae regios ad to use limits of summatios to fid areas of plae regios Let f be cotiuous ad oegative o the iterval [a, b]. The area A of the regio bouded b the graph of f, the -ais, ad the vertical lies = a ad = b is give b A = lim Σ f (a + (b a)i / )[(b a)/] Eample : Fid the area of the regio bouded b the graph + of f ( ) = ( 4) 5 ad the -ais betwee = 3 ad = 6. 8 Homework Assigmet Page(s) Eercises Larso/Hostetler/Edwards Precalculus with Limits: A Graphig Approach, Fifth Editio Studet Notetakig Guide IAE Copright Houghto Miffli Compa. All rights reserved.
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